Subgroups — C₃×#70707 #11790527

Order 1152.

421 conjugacy classes of subgroups.

The subgroup lattice has 279 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×#108	C₂	1	1	yes	maximal, characteristic
2	C₃×#5125	C₂	1	2	yes	maximal
3	C₂×C₃×#819	C₂	1	1	yes	maximal, characteristic
4	C₃×#5101	C₂	1	1	yes	maximal, characteristic
5	C₃×#4559	C₂	1	2	yes	maximal
6	C₃×#1953	—	3	1	no	maximal
7	#70707	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂×C₃²×M₁₆	C₂²	1	2	yes	—
9	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
10	C₃×#926	C₂²	1	1	yes	characteristic
11	C₂×C₃×#208	C₂²	1	2	yes	—
12	C₂×C₃×#208	C₂²	1	1	yes	characteristic
13	C₃²×#108	C₄	1	4	yes	complemented
14	C₃×#819	—	2	1	no	—
15	C₃×#819	—	2	1	no	—
16	C₃×#819	—	2	4	no	—
17	C₃×#815	—	2	1	no	—
18	C₃×C₈×Dic₃	—	2	1	no	—
19	C₃×#430	—	3	2	no	—
20	C₂×C₃×#108	—	2	1	no	—
21	C₂×C₃×#108	—	3	1	no	—
22	C₂×C₃×#108	D₃	1	1	yes	characteristic
23	C₂×C₃×#108	C₂×C₃	1	1	yes	characteristic
24	C₃×#405	—	3	1	no	—
25	#5125	C₂×C₃	1	2	yes	—
26	C₂×#819	C₂×C₃	1	1	yes	characteristic
27	#5101	C₂×C₃	1	1	yes	characteristic
28	C₃×#344	—	3	2	no	—
29	#4559	C₂×C₃	1	2	yes	—
30	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
31	C₃²×M₁₆	—	2	2	no	—
32	C₃²×M₁₆	C₂×C₄	1	4	yes	—
33	C₂×C₃²×C₈	—	2	2	no	—
34	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
35	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
36	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
37	C₂×C₃²×C₈	Q₈	1	2	yes	—
38	C₃×#217	Q₈	1	2	yes	—
39	C₃×C₄⋊Dic₃	—	2	1	no	—
40	C₃×C₄×Dic₃	—	2	1	no	—
41	C₃×#208	—	2	1	no	—
42	C₃×#208	—	2	2	no	—
43	C₃×#208	—	2	2	no	—
44	C₃×#208	D₄	1	2	yes	—
45	C₂×C₃×C₃⋊C₈	—	2	1	no	—
46	C₂×C₃×C₃⋊C₈	—	2	2	no	—
47	#1953	—	3	1	no	Sylow
48	C₂×C₃×M₁₆	—	2	2	no	—
49	C₂×C₃×M₁₆	—	3	1	no	—
50	C₂×C₃×M₁₆	—	3	2	no	—
51	C₂×C₃×M₁₆	C₂×D₃	1	2	yes	—
52	C₂×C₃×M₁₆	C₂²×C₃	1	2	yes	—
53	C₂²×C₃×C₈	—	2	1	no	—
54	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
55	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
56	C₃×#117	—	3	1	no	—
57	#926	C₂²×C₃	1	1	yes	characteristic
58	C₂×#208	C₂²×C₃	1	2	yes	—
59	C₂×#208	C₂²×C₃	1	1	yes	characteristic
60	C₃×#108	—	2	4	no	—
61	C₃×#108	—	6	4	no	—
62	C₃×#108	—	6	1	no	—
63	C₃×#108	—	6	1	no	—
64	C₃×#108	Dic₃	1	4	yes	complemented
65	C₃×#108	C₃×C₄	1	4	yes	complemented
66	C₃×#97	—	6	1	no	—
67	C₃×C₄×C₈	—	6	1	no	—
68	#819	—	2	1	no	—
69	#819	—	2	1	no	—
70	#819	—	2	4	no	—
71	#815	—	2	1	no	—
72	C₈×Dic₃	—	2	1	no	—
73	C₂³×C₃²	C₂×Q₈	1	1	yes	characteristic
74	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
76	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
77	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
78	C₂×C₃²×C₄	#41	1	1	yes	characteristic
79	C₂×C₃²×C₄	#41	1	1	yes	characteristic
80	C₂×C₃×Dic₃	—	2	2	no	—
81	C₃²×C₈	—	2	4	no	—
82	C₃²×C₈	C₄⋊C₄	1	2	yes	—
83	C₃²×C₈	C₄⋊C₄	1	2	yes	—
84	C₃×C₃⋊C₈	—	2	2	no	—
85	C₃×C₃⋊C₈	—	4	2	no	—
86	#430	—	3	2	no	—
87	C₂×#108	—	3	1	no	—
88	C₂×#108	C₃×D₃	1	1	yes	characteristic
89	#405	—	3	1	no	—
90	#344	—	3	2	no	—
91	C₂²×C₃×C₄	—	2	1	no	—
92	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
93	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
94	C₃×M₁₆	—	2	2	no	—
95	C₃×M₁₆	—	2	2	no	—
96	C₃×M₁₆	—	2	4	no	—
97	C₃×M₁₆	—	3	2	no	—
98	C₃×M₁₆	—	4	2	no	—
99	C₃×M₁₆	—	6	1	no	—
100	C₃×M₁₆	—	6	2	no	—
101	C₃×M₁₆	—	6	2	no	—
102	C₃×M₁₆	C₂×Dic₃	1	4	yes	—
103	C₃×M₁₆	C₂×C₃×C₄	1	4	yes	—
104	C₂×C₃×C₈	—	2	2	no	—
105	C₂×C₃×C₈	—	2	2	no	—
106	C₂×C₃×C₈	—	2	2	no	—
107	C₂×C₃×C₈	—	2	1	no	—
108	C₂×C₃×C₈	—	2	1	no	—
109	C₂×C₃×C₈	—	2	2	no	—
110	C₂×C₃×C₈	—	4	2	no	—
111	C₂×C₃×C₈	—	6	1	no	—
112	C₂×C₃×C₈	—	6	2	no	—
113	C₂×C₃×C₈	Dic₆	1	2	yes	—
114	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
115	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
116	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
117	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
118	C₂×C₃×C₈	C₃×D₄	1	1	yes	characteristic
119	C₂×C₃×C₈	C₃×D₄	1	1	yes	characteristic
120	C₂×C₃×C₈	C₃×Q₈	1	2	yes	—
121	C₃×C₄⋊C₄	—	6	1	no	—
122	C₃×C₂²⋊C₄	—	3	2	no	—
123	C₃×C₄²	—	6	1	no	—
124	#217	C₃×Q₈	1	2	yes	—
125	C₄⋊Dic₃	—	2	1	no	—
126	C₄×Dic₃	—	2	1	no	—
127	#208	—	2	1	no	—
128	#208	—	2	2	no	—
129	#208	—	2	2	no	—
130	#208	C₃×D₄	1	2	yes	—
131	C₂×C₃⋊C₈	—	2	1	no	—
132	C₂×C₃⋊C₈	—	2	2	no	—
133	C₂²×C₃²	—	4	1	no	—
134	C₂²×C₃²	C₄×Q₈	1	1	yes	characteristic
135	C₂²×C₃²	C₂²⋊Q₈	1	1	yes	characteristic
136	C₂²×C₃²	C₄⋊Q₈	1	1	yes	characteristic
137	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
138	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
139	C₃²×C₄	#121	1	1	yes	characteristic
140	C₃²×C₄	#125	1	1	yes	characteristic
141	C₃×Dic₃	—	4	2	no	—
142	C₂×M₁₆	—	3	1	no	—
143	C₂×M₁₆	—	3	2	no	—
144	C₂×M₁₆	C₂×C₃×D₃	1	2	yes	—
145	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
146	#117	—	3	1	no	—
147	#108	—	6	4	no	—
148	#108	—	6	1	no	—
149	#108	—	6	1	no	—
150	#108	C₃×Dic₃	1	4	yes	complemented
151	#97	—	6	1	no	—
152	C₄×C₈	—	6	1	no	—
153	C₂³×C₃	—	2	1	no	—
154	C₂³×C₃	D₃×Q₈	1	1	yes	characteristic
155	C₂³×C₃	C₂×C₃×Q₈	1	1	yes	characteristic
156	C₂×C₃×C₄	—	2	1	no	—
157	C₂×C₃×C₄	—	2	1	no	—
158	C₂×C₃×C₄	—	2	1	no	—
159	C₂×C₃×C₄	—	2	1	no	—
160	C₂×C₃×C₄	—	2	1	no	—
161	C₂×C₃×C₄	—	2	1	no	—
162	C₂×C₃×C₄	—	6	2	no	—
163	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
164	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
165	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
166	C₂×C₃×C₄	#237	1	1	yes	characteristic
167	C₂×C₃×C₄	#239	1	1	yes	characteristic
168	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
169	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
170	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
171	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
172	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
173	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
174	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
175	C₂×Dic₃	—	2	2	no	—
176	C₃×C₈	—	2	4	no	—
177	C₃×C₈	—	2	4	no	—
178	C₃×C₈	—	2	2	no	—
179	C₃×C₈	—	2	2	no	—
180	C₃×C₈	—	4	4	no	—
181	C₃×C₈	—	6	2	no	—
182	C₃×C₈	—	12	2	no	—
183	C₃×C₈	C₄⋊Dic₃	1	2	yes	—
184	C₃×C₈	C₄⋊Dic₃	1	2	yes	—
185	C₃×C₈	C₃×C₄⋊C₄	1	2	yes	—
186	C₃×C₈	C₃×C₄⋊C₄	1	2	yes	—
187	C₃⋊C₈	—	2	2	no	—
188	C₃⋊C₈	—	4	2	no	—
189	C₂×C₃²	—	2	1	no	—
190	C₂×C₃²	—	2	1	no	—
191	C₂×C₃²	—	2	1	no	—
192	C₂×C₃²	#389	1	1	yes	characteristic
193	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
194	M₁₆	—	2	2	no	—
195	M₁₆	—	3	2	no	—
196	M₁₆	—	6	1	no	—
197	M₁₆	—	6	2	no	—
198	M₁₆	—	6	2	no	—
199	M₁₆	C₂×C₃×Dic₃	1	4	yes	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	6	1	no	—
202	C₂×C₈	—	6	2	no	—
203	C₂×C₈	C₃×Dic₆	1	2	yes	—
204	C₂×C₈	C₃×D₁₂	1	1	yes	characteristic
205	C₂×C₈	C₃×D₁₂	1	1	yes	characteristic
206	C₂×C₈	C₂×C₃×Dic₃	1	2	yes	—
207	C₄⋊C₄	—	6	1	no	—
208	C₂²⋊C₄	—	3	2	no	—
209	C₄²	—	6	1	no	—
210	C₂²×C₃	—	2	1	no	—
211	C₂²×C₃	—	2	1	no	—
212	C₂²×C₃	—	2	1	no	—
213	C₂²×C₃	—	4	1	no	—
214	C₂²×C₃	—	4	1	no	—
215	C₂²×C₃	—	4	2	no	—
216	C₂²×C₃	#888	1	1	yes	characteristic
217	C₂²×C₃	#897	1	1	yes	characteristic
218	C₂²×C₃	Q₈×Dic₃	1	1	yes	characteristic
219	C₂²×C₃	C₃×C₄×Q₈	1	1	yes	characteristic
220	C₂²×C₃	C₃×C₂²⋊Q₈	1	1	yes	characteristic
221	C₂²×C₃	C₃×C₄⋊Q₈	1	1	yes	characteristic
222	C₃×C₄	—	2	1	no	—
223	C₃×C₄	—	2	1	no	—
224	C₃×C₄	—	2	1	no	—
225	C₃×C₄	—	2	1	no	—
226	C₃×C₄	—	12	2	no	—
227	C₃×C₄	#890	1	1	yes	characteristic
228	C₃×C₄	#895	1	1	yes	characteristic
229	C₃×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
230	C₃×C₄	D₄×Dic₃	1	1	yes	characteristic
231	C₃×C₄	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
232	C₃×C₄	C₃×C₄×D₄	1	1	yes	characteristic
233	C₃×C₄	C₃×#121	1	1	yes	characteristic
234	C₃×C₄	C₃×#125	1	1	yes	characteristic
235	Dic₃	—	4	2	no	—
236	C₃²	#1953	1	1	yes	characteristic, complemented, Sylow
237	C₂³	C₃×D₃×Q₈	1	1	yes	characteristic
238	C₂×C₄	—	6	2	no	—
239	C₂×C₄	C₂×C₃×Dic₆	1	1	yes	characteristic
240	C₂×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
241	C₂×C₄	C₃×D₃×D₄	1	1	yes	characteristic
242	C₂×C₄	C₃×#237	1	1	yes	characteristic
243	C₂×C₄	C₃×#239	1	1	yes	characteristic
244	C₂×C₄	C₂²×C₃×Dic₃	1	1	yes	characteristic
245	C₈	—	2	4	no	—
246	C₈	—	6	2	no	—
247	C₈	—	12	2	no	—
248	C₈	C₃×C₄⋊Dic₃	1	2	yes	—
249	C₈	C₃×C₄⋊Dic₃	1	2	yes	—
250	C₂×C₃	—	2	2	no	—
251	C₂×C₃	—	2	1	no	—
252	C₂×C₃	—	2	1	no	—
253	C₂×C₃	—	2	1	no	—
254	C₂×C₃	—	2	1	no	—
255	C₂×C₃	—	2	1	no	—
256	C₂×C₃	—	2	1	no	—
257	C₂×C₃	—	2	1	no	—
258	C₂×C₃	—	4	1	no	—
259	C₂×C₃	—	4	1	no	—
260	C₂×C₃	#4984	1	1	yes	characteristic
261	C₂×C₃	C₃×#389	1	1	yes	characteristic
262	C₂²	—	4	1	no	—
263	C₂²	C₃×#888	1	1	yes	characteristic
264	C₂²	C₃×#897	1	1	yes	characteristic
265	C₂²	C₃×Q₈×Dic₃	1	1	yes	characteristic
266	C₄	—	12	2	no	—
267	C₄	C₃×#890	1	1	yes	characteristic
268	C₄	C₃×#895	1	1	yes	characteristic
269	C₄	C₂×C₃×C₄⋊Dic₃	1	1	yes	characteristic
270	C₄	C₃×D₄×Dic₃	1	1	yes	characteristic
271	C₃	—	2	1	no	—
272	C₃	#70707	1	1	yes	characteristic, direct product factor
273	C₃	C₃×#1953	1	1	yes	characteristic, complemented
274	C₂	—	2	1	no	—
275	C₂	—	2	1	no	—
276	C₂	—	2	1	no	—
277	C₂	C₃×#4984	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	#895
commutator	C₂×C₃×C₄	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₃
Fitting	C₂×C₃²×#108	C₂
socle	C₂×C₃²	#389
prime radical	C₂³×C₃²	C₂×Q₈
cosocle	C₂²×C₃×C₄	C₂³×C₃
hypercenter	C₂×C₃×#108	D₃
hypocenter	C₃	C₃×#1953
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₃×C₄ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×#108

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
7. #70707	C₃	direct (N × H)
13. C₃²×#108	C₄	semidirect
64. C₃×#108	Dic₃	semidirect
65. C₃×#108	C₃×C₄	semidirect
150. #108	C₃×Dic₃	semidirect
236. C₃²	#1953	semidirect
272. C₃	#70707	direct (N × H)
273. C₃	C₃×#1953	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃²×#108	2
C₃×#5125	2
C₂×C₃×#819	2
C₃×#5101	2
C₃×#4559	2
C₃×#1953	3
#70707	3

Subgroups — C3×#70707 #11790527

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#70707 #11790527